PREFACE

Gui-Qiang G.CHEN Bo LI Zizhou TANG Xiping ZHU

This special issue of Acta Mathematica Scientia is dedicated to Professor Banghe Li on the occasion of his 80th birthday.

Professor Banghe Li was born in Yueqing City,Zhejiang Province.He graduated from the University of Science and Technology of China in 1965 and has been working in the Chinese Academy of Sciences since then.He was elected as an Academician of the Chinese Academy of Sciences in 2001.

Professor Lihasdevoted great effort to the unity of Mathematics,and has made systematic contributions in several fields.He has published 2 books and over 100 research papers.His main research fields are Differential Topology and Low-Dimensional Topology.

On Immersion Theory,a major subject in Differential Topology,he established new frameworksand techniques,corrected the errors of some famous mathematicians,and obtained many results that opened up new aspects of this subject.One of the two most fundamental theorems in Differential Topology states that any n-manifold can be immersed in(2n?1)-Euclidean spaces.This theorem was generalized by Professor Li and Professor Peterson from the simplest manifolds(Euclidean spaces)to arbitrary manifolds in the following form:any map from an n-manifold to a(2n?1)-manifold is homotopic to immersions.Professor Li also solved the problem of classifying immersions for all maps from n-manifolds to 2n-manifolds,and those from 2-manifolds to 3-manifolds.Furthermore,he(with his collaborators)solved the diffi-cult problems of classifying immersions from n-manifolds into(2n?2)-Euclidean space,and classifying immersions from k-connected n-manifolds into(2n?k)-Euclidean space.

In the field of Low-Dimensional Topology,one of the typical problems in 4-dimensional topology is to represent a homology class in a 4-dimensional manifold by spheres.Although many famous mathematicians(such as Milnor)tried to solve this problem over the course of 40 years,it remains a difficult problem.The mathematical theory of gauge fields did bring breakthroughs;so far,for 8 simply connected 4-manifolds,the problem has been solved completely.Professor Li has contributed five of these solutions.For the minimal genus problem in 4-dimensional topology,he created several positive constructive techniques that brought about important breakthroughs and were regarded as a“dream”by Donaldson(Fields medalist).In studying the Witten invariants of 3-dimensional manifolds,Professor Li found new invariants,cleared the relation of various invariants,completely solved the problem of generalized Gauss sums in Algebraic Number Theory,calculated the Witten invariants for all lens spaces,and answered the problem posed by Kirby.

Professor Lihasalso madecontributions in fields such as Nonstandard Analysis,Generalized Functions,Partial Differential Equations,and Mathematical Biology.

About 20 years ago,Professor Li started to study Hilbert’s 15th Problem,which is one of the few Hilbert’s problems that remains unsolved.He found that Nonstandard Analysis is a suitable tool for addressing this problem.The problem falls within the scope of Enumerative Geometry,which deals with finding the number of points,lines,and planes that satisfy some specified conditions.However,as was often the casein Algebraic Geometry more than a century ago,the methods of Enumerative Geometry were intuitive and not rigorous.From a modern point of view,this problem belongs to the field of Intersection Theory.The key to the problem is to understand Schubert’s book on Enumerative Geometry.Schubert’s book“Kalkülder Abz?hlenden Geometrie”was written in German,and there was no other translation at that time.Professor Li invited Professor Peilian Li to translate Schubert’s book in German into Chinese in 2005.After that,Professor Li took about 10 years to understand the basic structure of the book,and finally he and Professor Jianmin Yu proofread Professor Peilian Li’s transcript.That took them about 3 years.The Chinese edition of Schubert’s book was published in 2018.

One of Schubert’s main contributions was using a technique of reducing to degenerate cases.When Schubert used this technique of reducing to degenerate cases,he actually thought of the process by which normal geometrical objects move infinitely close to some degenerate figures.It is not difficult to see that Schubert used the variety consisting of geometrical objects regarded in terms of its points,and took a point in the variety that represents a degenerate case which is very near to a point representing special objects satisfying certain conditions.The common way to define the word“near”is to use“epsilon-delta”language in a metric space.However,it is very difficult to deal with complicated problems with so many different reduced degenerate cases.Fortunately,the tool for understanding this complicated situation concerning infinitesimals was established in the 1960s;the tool is Nonstandard Analysis.Professor Li gave a rigorous proof for Example 4 in Section 4 of Schubert’s book according to Schubert’s idea.In this example,Schubert used one line with multiplicity m and n points on the line to represent a degenerate curve with order m and rank n.This idea was intuitive but not rigorous.

Multiplicity is another important drawback of Schubert’s work.In the era of Schubert,the rigorous definition of multiplicity was not established.He determined multiplicity according to intuition and experience.It seemed that he was not so confident about the rigor of the multiplicity,as he always used different methods to check his formulas and numbers to show that his method was reliable.Nowadays,the definition of multiplicity is clear.To calculate the multiplicity,we need to write down the equations of the relevant varieties.However,in Schubert’s book,there are no equations of varieties at all.Fortunately,the Ritt-Wu method and the Gr?bner bases are now available to let computers do the calculations.Professor Li proved a statement in Section 23 of Schubert’s book by using these methods.

In addition,Professor Li systematically studied some of the basic formulas in Schubert’s book;these included incidence formulas and coincidence formulas.

The many important contributions made by Professor Li have won him a reputation around the world.He won the Shiing Shen Chern Mathematics Award and the Hua Loo-Keng Prize of Mathematics from the Chinese Mathematical Society in 1989 and 2009,respectively.He was honored with the Prize for Scientific and Technological Achievements by the Ho Leung Ho Lee Foundation in Mathematics and Mechanics in 2010.

Professor Banghe Li’s distinguished achievements in Mathematics and his passion for the field have profoundly influenced us.The aim of this issue is to collect some up-to-date research papers that are closely related to his work and have been contributed by his collaborators,former students,postdoctoral fellows,and friends.

On this special occasion of his 80th birthday,we wish Professor Banghe Li the very best of health and the best of wishes for his forthcoming mathematical endeavors.